Gauge field fermion Lagrangian

As in the case of the electromagnetic self-mass, the implied dynamical mass increment is infinite unless perturbation-theory sums are truncated by a renormalization cutoff procedure. In analogy to electrodynamics, each fermion field acquires an incremental dynamical mass through interaction with the gauge field. This implies in electroweak theory that neutrinos must acquire such a dynamical mass from their interaction with the SUIT) gauge field. For a renormalized Dirac fermion in an externally determined SUIT) gauge field, the Lagrangian density is... 

In quantum field theory, the gauge field is determined by its Lagrangian density, and the fermion field, by the Dirac Lagrangian density ... 

The total Lagrangian X = JS G + JS D + JS , then involves the interaction between fermions and the gauge field. The Dirac field will be generically considered to be the electron and the gauge theory will be considered to be the non-Abelian electromagnetic field. The theory then describes the interaction between electrons and photons. A gauge theory involves the conveyance of momentum form one particle (electron) to another by the virtual creation and destruction of a vector boson (photon) that couples to the two electrons. The process can be diagrammatically represented as... 

In the presence of an externally determined fermion gauge current, the Lagrangian density for the 5(7(2) gauge field is [435]... 

The Lagrangian density for a massless fermion field interacting with the S(J(2) gauge field is... 

Because field quantization falls outside the scope of the present text, the discussion here has been limited to properties of classical fields that follow from Lorentz and general nonabelian gauge invariance of the Lagrangian densities. Treating the interacting fermion field as a classical field allows derivation of symmetry properties and of conservation laws, but is necessarily restricted to a theory of an isolated single particle. When this is extended by field quantization, so that the field amplitude rjr becomes a sum of fermion annihilation operators, the theory becomes applicable to the real world of many fermions and of physical antiparticles, while many qualitative implications of classical gauge field theory remain valid. 

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